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MPP-2010-159 PreprinttypesetinJHEPstyle-HYPERVERSION DIAS-STP-10-13 Mesons from global Anti-de Sitter space Johanna Erdmenger and Veselin Filev ∗ Max-Planck-Institut fu¨r Physik (Werner-Heisenberg-Institut) Föhringer Ring 6, 80805 Mu¨nchen, Germany Abstract: In the context of gauge/gravity duality, we study both probe D7– and probe D5–branes in global Anti-de Sitter space. The dual field theory is N = 4 theory on R×S3 with added flavour. The branes undergo a geometrical phase transition in this geometry as function of the bare quark mass m in units of 1/R with R the S3 q radius. Themesonspectraareobtainedfromfluctuationsofthebraneprobes. First, we study them numerically for finite quark mass through the phase transition. Moreover, at zero quark mass we calculate the meson spectra analytically both in supergravity and in free field theory on R × S3 and find that the results match: For the chiral primaries, the lowest level is given by the zero point energy or by the scaling dimension of the operator corresponding to the fluctuations, respectively. The higher levels are equidistant. Similar results apply to the descendents. Our results confirm the physical interpretation that the mesons cannot pair-produce any further when their zero-point energy exceeds their binding energy. Keywords: AdS/CFT correspondence, Gauge/gravity correspondence. ∗E-mail addresses: jke, filev @mppmu.mpg.de Contents 1. Introduction 2 2. General Setup 5 2.1 Probe D7–brane. 5 2.2 Probe D5–brane 8 3. Meson spectra 10 3.1 Fluctuations of the D7–brane embedding. 11 3.1.1 Fluctuations of the transverse scalars. 11 3.2 Fluctuations of the D5–brane embedding. 15 4. Spectrum at zero bare mass – Gravity side 18 4.1 Fluctuations of a D7–brane probe 19 4.1.1 Fluctuations along L 19 4.1.2 Fluctuations along φ. 20 4.1.3 Fluctuations of the gauge field. 21 4.2 Fluctuations of a D5–brane probe 28 4.2.1 Fluctuations of l(r) 28 4.2.2 Fluctuation of the Neumann-Dirichlet coordinate. 29 5. Spectrum at zero bare mass – Field theory side 31 5.1 A Conformally coupled scalar on R1 ×Sñ 31 5.2 A collection of fields on S˜3 32 5.3 A collection of fields on S˜2 37 6. Conclusion 40 7. Acknowledgements 41 – 1 – 1. Introduction The AdS/CFT correspondence in its original form involves the near-horizon limit of D3–branes. This limit leads to the AdS ×S5 geometry, where the Anti-de Sitter factor 5 of the geometry is realized as the Poincaré patch of AdS . 5 Following[1],therehavebeenextensivestudiesofprobeD7–branesinthisgeometry which on the dual field theory side leads to added hypermultiplets in the fundamental representation of the gauge group. For preserving N = 2 supersymmetry, the probe D7–branehastowrapasubspacewhichisasymptoticallyAdS ×S3 neartheboundary. 5 The complete meson spectra arising from the fluctuations of the probe brane, ie. for all scalar, fermion and vector modes, have been found in [2]. For instance, for scalar mesons the result is [2] M (n,l) = 2L(cid:112)(n+l+1)(n+l+2), L = √2π√mq , (1.1) s R2 R2 λ where L is the embedding coordinate of the D7 brane, R is the AdS radius, m is q the quark mass and λ the ’t Hooft coupling. n is the main quantum number and l is the quantum number of the SO(4) symmetry associated with the S3 asymptotically wrapped by the D7–brane probe near the boundary. The spectrum has a large degener- acy since it depends only on the combination (n+l). This is expected from the N = 2 supersymmetry of the system. For general Dp/Dq–brane systems, similar spectra were found in ref. [3, 4]. By embedding a D7–brane probe into deformed versions of AdS × S5, physical 5 phenomena may be described. Examples are chiral symmetry breaking, which is obtained, by embedding a D7–probe into a confining background [5], and a first-order meson-melting phase transition at finite temperature which arises when embedding a probe brane into the AdS-Schwarzschild black hole geometry [5, 6, 7, 8]. For the AdS-Schwarzschild geometry, a topology-changing phase transition occurs, depending on the ratio of quark mass over temperature, between branes that reach the black hole horizon and those that do not. Supersymmetric embeddings of D5–brane probes wrapping an AdS ×S2 of AdS × 4 5 S5 have first been studied in [9]. In this case, the dual field theory is a superconformal ‘defect’ theory [10] in which the additional hypermultiplets are confined to a (2+1)- dimensional subspace. In [11] (see also [12]), the authors investigate the embedding of D7– and D5–brane probes into global AdS. Global AdS is dual to N = 4 Super Yang-Mills theory on R×S3. Moreover these authors study the thermodynamical properties of these systems by considering brane probes in global thermal AdS, where also the time direction is – 2 – compactified, such that the field theory is defined on S1×S3. Again in these geometries they find a topology-changing phase transition, depending on the ratio of quark mass over S3 radius, between those embeddings that reach the S3 and those that do not. Using critical exponents, the authors show (see also refs. [13, 14, 15]) that the phase transitionisthirdorderforD7–braneprobes, whileitisfirstorderforD5–braneprobes. They also consider fluctuations of the brane and calculate the mass of the lowest-lying scalar meson mode. The authors find that the spectrum has a kink at the phase transition. Moreover, by making use of the Polyakov loop, they study the physical properties of the low-energy phase. In this phase, the zero-point energy of the mesons – which is due to the finite volume of the S3 – is larger than their binding energy. This means in particular that the mesons are deconfined in the sense that they cannot pair-produce any more. String breaking is no longer possible. In this paper we investigate the mesonic spectrum on R×S3 in further detail. We look at scalar and vector fluctuations for both the D7– and the D5–brane probe cases. For the D7 brane probe, we establish the complete bosonic spectrum on the gravity side. Our most important result concerns the spectrum at vanishing quark mass, for which we perform analytical calculations both on the gravity and on the field theory side. On the gravity side, we map the fluctuation equations of motion to equations of Schro¨dingertype. Forthechiralprimaries, wefindthattheenergyofthegroundstateis given by the dimension of the operator dual to the fluctuations. The higher fluctuation modes, labelled by the quantum number n, are equidistant. For the descendents, we also find an equidistant spectrum. However the energy of the ground state is no longer equal to the dimension of the operator. We then turn to the field theory side, and in particular to the free N = 2 theory in 3+1 dimensions obtained by adding a flavour hypermultiplet to the original N = 4 theory. We expand the fields on the S3 within R×S3. The spectrum of the mesonic composite operators is obtained by combining the expansions of the component fields, making use of the appropriate Clebsch-Gordan coefficients. This requires a careful analysis of the transformation properties of the mesonic composite operators under the antipodal map.1 For both the chiral primaries and the descendents, we find the same result for the spectrum as in the gravity calculation. As an example, let us quote our result for scalar D7–brane mesons, which is 1 ˜ ˜ M(n,l,l) = (3+2n+l+l). (1.2) R Here, R is the radius of the S3 in the field theory directions, which gives rise to the 1The antipodal map A : Sn → Sn, defined by A(x) = −x, sends every point on the sphere to its antipodal point. – 3 – conformalmass1/R. Notethatsincethismassarisesfromthebackgroundmetricrather than from the D7–brane boundary conditions, (1.2) is independent of the ’t Hooft √ coupling λ, while (1.1) is O(1/ λ). Moreover, (1.2) depends on the main quantum number n, as well as on the S3 quantum numbers l and ˜l, where l refers to the internal S3, asymptotically wrapped by the D7–brane, while ˜l refers to the S3 in R × S3 in the field theory directions. The lowest mode, ie. the zero-point energy, is given by the conformal dimension of the dual operator, ∆ = 3. It is remarkable that the free field calculation and the gravity calculation of the meson spectrum agree. This is of course due to non-renormalization theorems which hold also in N = 2 theory. From the physics perspective this confirms the physical interpretation, already advocated in [11], that the mesons cannot pair-produce even at strong coupling when confined into a small volume such that their zero-point energy is larger than their binding energy. ˜ We note that when setting l to zero, our new result (1.2), which in this case depends on the combination 2n + l, is less degenerate than (1.1) which depends on n + l. This linked to the fact that for supersymmetric field theories on S3, scalars, fermions and vectors in the same multiplet have different conformal mass. Also, the supersymmetry transformations of the field theory fermions are modified to contain curvature-dependent terms. For N = 4 Super-Yang-Mills theory on R × S3, these issues have been studied in detail in [16]. We expect similar results to hold for the N = 2 theory considered here. Our calculation shows - in a simple example - that it is possible to directly compare gravity and field theory calculations in top-down approaches to holographic models of physicalrelevance. Suchtop-downmodelshavebeenusedwidelyrecentlytoholograph- ically describe physical phenomena such as superfluidity, quantum critical points and transport properties [17, 18, 19, 20]. These models have the advantage - as compared to bottom-up models - that the dual field theory is explicitly known. We hope that further field-theory studies will follow soon for a more detailed comparison between the weak and strong coupling aspects of a given model. This paper is organized as follows. In section 2 we introduce the general setup of global AdS and its probe brane embeddings, and present the phase transitions for the D7 and D5 brane cases. In section 3 we study the probe brane fluctuations as function of the bare quark mass and discuss the behaviour of the meson spectrum for both the phase transition and the case of vanishing quark mass. In section 4, we analytically compute the meson spectra at zero quark mass on the gravity side. We obtain the full bosonic spectrum for the D7 brane fluctuations, as well as some characteristic examples for the D5 brane case. In section 5 we present the field theory calculation at zero quark mass and show that it agrees with the gravity results. We end with concluding remarks – 4 – in section 6. 2. General Setup Our starting point is AdS ×S5 in global coordinates: 5 dw2 ds2 = −(1+w2/R2)dt2 +w2dΩ2 + +R2dΩ2 . (2.1) 3 1+w2/R2 5 It is convenient to introduce the following radial coordinate: 1 √ u = (w+ R2 +w2) . (2.2) 2 The metric (2.1) in these coordinates is given by: u2 (cid:18) R2 (cid:19)2 u2 (cid:18) R2 (cid:19)2 R2 (cid:0) (cid:1) ds2 = − 1+ dt2 + 1− dΩ2 + du2 +u2dΩ2 . (2.3) R2 4u2 R2 4u2 3 u2 5 Note that u ≥ R/2. In this way the transverse R6 has a ball of radius R/2 sitting at the origin. 2.1 Probe D7–brane. Let us introduce a probe D7–brane. To this end it is convenient to write the metric of unit S5 in the following coordinates: ds2 = dθ2 +cos2θdΩ2 +sin2θdφ2 . (2.4) S5 3 Now if we let the D7-brane be extended along the AdS part of the geometry and wrap 5 an S˜3 ⊂ S5, the radial part of the corresponding DBI lagrangian is given by: (cid:18) R4 (cid:19)(cid:18) R2 (cid:19)2 (cid:112) L ∝ 1− 1− u3cos3θ 1+u2θ(cid:48)(u)2 . (2.5) 16u4 4u2 Possible embeddings split into two classes Minkowski embeddings and “ball” embeddings correspondingly wrapping shrinking S3 cycles in the S5 and AdS parts of the 5 background [12] (look at figure 1). The two classes are separated by a critical embedding which has conical singularity at the ball (represented by the dashed curve in figure 1). According to the AdS/CFT dictionary the bulk dynamics of the scalar θ(u) encodes the dynamics of the dual gauge invariant operator. In particular one can read off the source and the vev of the operator from the asymptotic behaviour of θ(u). In – 5 – u sinΘ 1.4 1.2 1.0 0.8 0.6 0.4 0.2 u cosΘ 0.0 0.5 1.0 1.5 2.0 Figure 1: Blue curves correspond to Minkowski embeddings and Red curves correspond to “ball” embeddings. The black dashed line corresponds to the critical embedding. our case the operator is the fundamental bilinear and the source and vev of the operator correspond to the mass of the hypermultiplet and the fundamental condensate. The precise dictionary has been derived in ref. [21], where an elegant renormalization prescription has been offered. Let us briefly review the results of refs. [21, 12] in a slightly modified form relevant to our notations. For large u the solution to the equation of motion derived from (2.5) has the following expansion: θ R θ R3 θ R3 u 0 2 0 θ(u) = + − log +... . (2.6) u u3 2u3 R After following the renormalization prescription outlined in ref. [21] the condensate of the theory can be calculated in terms of the parameters θ ,θ . For our choice of radial 0 2 coordinate the result is given by: θ3 (cid:104)ψ¯ψ(cid:105) ∝ −2θ + 0 +θ logθ ; . (2.7) 2 0 0 3 If we split the R6 space corresponding to du2 + u2dΩ2 to R4 × R2 and define radial 5 coordinates ρ = ucosθ and L = usinθ the DBI lagrangian in this coordinates is given by: (cid:18) R2 (cid:19)(cid:18) R2 (cid:19)3 √ L ∝ 1+ 1− ρ3 1+L(cid:48)2 . (2.8) 4u2 4u2 – 6 – TheprofileoftheD7–braneembeddingL(ρ)hasthefollowingasymptoticbehaviour at large ρ: c m 1 L(ρ) = m+ − logρ+... . (2.9) ρ2 2ρ2 Further more one has the relations m = θ R and c = R3(θ −θ3/6). The condensate 0 1 2 0 of the theory is then given by: (cid:104)ψ¯ψ(cid:105) ∝ −2c +mR2log(m/R) ≡ −2c , (2.10) 1 where we have defined a new parameter c proportional to the condensate. Note also that the bare mass of the hypermultiplet is proportional to m the exact relation is m = m/2πα(cid:48). q After solving numerically for the D7–brane embeddings we can generate a plot of the equation of state c(m) presented in figure 2, where we have used dimensionless parameters m˜ = m/R and c˜= c/R3. As one can see there is no apparent multi-valued (cid:142) (cid:45)c (cid:142) m 0.5 1.0 1.5 2.0 (cid:45)0.05 (cid:45)0.10 (cid:45)0.15 Figure 2: A plot of the condensate −c˜versus the bare mass m˜. The states corresponding to “ball” embeddings are presented by the red curve and the states corresponding to Minkowski embeddings are presented by the blue curve. region near the transition from Minkowski to “ball” embeddings. In fact it has been shown that the phase transition is continuous and is of a third order [11]. Let us comment briefly on the physical meaning of the parameter m˜ = m/R and in particular its R dependence. Consider a constant u = Λ (cid:29) R slice of the AdS ×S5 5 space-time. The induced metric has the asymptotic form: Λ2 ds2 = (dt2 +R2dΩ )+R2dΩ2 . (2.11) Λ R2 3 5 – 7 – Here Λ is specifying the energy scale of the dual gauge theory. It is natural to identify the radius of the AdS space-time R with the radius of the three sphere R where the 5 3 holographically dual field theory is defined. Furthermore we know that m = 2πα(cid:48)m , q where m is the bare mass of the hypermultiplet. However the quantity m /R is q q 3 obviously not a dimensionless parameter. To clarify this let us consider a rescaling Λ → γΛ of the energy scale. The metric in equation (2.11) becomes: Λ2 ds = (γ2dt2 +γ2R2dΩ2)+R2dΩ2. (2.12) γΛ R2 3 5 We can always scale the time coordinate t → t/γ to compensate for the rescaling of the energy scale. Note that this suggests that the bare mass of the hypermultiplet should also be rescaled m → γm . Furthermore the radius of the three sphere is now q q R = γR and hence we can write: γ = R /R. For the parameter m˜ we obtain: 3 3 γm (2πα(cid:48)) 2πα(cid:48) π m R m˜ = q = m R = √ √q 3 , (2.13) R q 3 R2 2 λ where we have used that R2 is related to the t’Hooft coupling of the dual gauge theory via: R2 = 2λα(cid:48). Equation (2.13) specifies the physical meaning of the parameter m˜. Now we can interpret the phase transition as taking place at constant bare mass m q and varying radius of S3. Note that varying R corresponds to varying the Casimir 3 energy of the dual gauge theory and hence it is essentially a quantum phase transition. 2.2 Probe D5–brane Let us review the case of a probe D5–brane studied in ref. [11]. To this end it is convenient to write the S5 part of the geometry in the following coordinates: ds2 = dψ2 +cos2ψdΩ2 +sin2ψdΩ˜2 . (2.14) S5 2 2 Next we define: r = ucosψ and l = usinψ. The metric (2.3) can then be written as: u2 (cid:18) R2 (cid:19)2 u2 (cid:18) R2 (cid:19)2 ds2 = − 1+ dt2 + 1− [dα2 +sin2α(dβ2 +sin2βdγ2)] R2 4u2 R2 4u2 R2 + [dr2 +r2dΩ2 +dl2 +l2dΩ˜2] . (2.15) u2 2 2 Now if we let the D5–brane be extended along the t,β,γ,r,Ω directions and has a 2 – 8 – non-trivial profile along l, the corresponding DBI lagrangian is given by: (cid:18) R2 (cid:19)(cid:18) R2 (cid:19)2 √ L ∝ 1+ 1− r2 1+l(cid:48)2 . (2.16) 4u2 4u2 It is easy to show that the solution to the equation of motion has the following asymptotic behaviour at large r: c l(r) = m+ +.... (2.17) r Note that unlike the D7–brane case there is no extra logarithmic term [21] in the asymptotic expansion of l(r). One can then directly relate the coefficient c to the fundamental condensate of the theory ((cid:104)q¯q(cid:105) ∝ −c). It is convenient to define dimensionless quantities c˜= c/R2 and m˜ = m/R. After solving numerically the equation of motion of the probe D5–brane we obtain the plot of −c˜ versus m˜ presented in figure 3. As one can see there is a multi–valued (cid:142) (cid:45)c 0.6 0.5 0.4 0.3 0.2 0.1 (cid:142) m 0.5 1.0 1.5 2.0 2.5 (cid:142) (cid:45)c 0.53 0.52 0.51 0.50 0.49 (cid:142) m 1.196 1.198 1.200 1.202 1.204 1.206 Figure 3: A plot of the condensate −c˜ versus the bare mass m˜ for the D5–brane probe. The states corresponding to “ball” embeddings are presented by red curves and the states corresponding to Minkowski embeddings are presented by blue curves. The zoomed in plot suggests a first order phase transition in the dual gauge theory. – 9 –